Poincar\'e Mechanism in Multi-scaled Hamiltonian Systems

My talk is about the quasi-periodic motions in multi-scaled Hamiltonian systems.
It consists of four part. At first, I will introduce the results in integrable Hamiltonian systems
since what we focus on is nearly-integrable Hamiltonian system. The second part is the definition of
nearly-integrable Hamiltonian system and the classical KAM theorem. After then, I will introduce that
what is Poincar\'e problem and some interesting results corresponding to this problem. The last part,
which is also the main part, I will talk about the definition and the background of nearly-integrable
Hamiltonian system, then the persistence of lower dimensional tori on resonant surface, which is our recent
result. I will also simply introduce the Technical ingredients of our work.